Michael Ward

UBC

Beyond Turing: The Stability and Dynamics of Localized Patterns in Reaction-Diffusion Systems

Since the pioneering work of Turing in 1952, there have been many
studies of instabilities of spatially homogeneous patterns in
reaction-diffusion systems. These previous results provide general
criteria for the onset of different types of instabilities as well as
providing a normal form analysis valid near the onset of the
instability. However, in the singularly perturbed limit, many
reaction-diffusion systems can give rise to solutions that have a high
degree of spatial heterogeneity. Examples of this class of solutions
include grain boundaries in materials science, vortices in
superconductivity, hot-spot solutions in the microwave heating of
ceramics, and spike-type solutions related to biological
morphogenesis. In contrast to spatially homogeneous solutions, the
instabilities and the dynamics of these localized patterns are not
nearly as well understood. In this talk, we begin by giving a brief
survey of problems and general results in pattern formation theory.
Then, we highlight novel types of dynamical behaviors that occur for
spike-type solutions to the Gierer-Meinhardt system modeling
morphogenesis. These behaviors include both self-replication
instabilities, whereby new spikes are created across the domain,
and intricate temporal instabilities initiated through a subcritical
Hopf bifurcation. These problems are studied using a combination of
rigorous, asymptotic, and numerical methods. A nonlocal eigenvalue
problem is found to be central to the stability analysis. Our results
are compared with recent results of Ni, Takagi, and Yanagida.